The frequency of vibration $f$ of a mass $m$ suspended from a spring of spring constant $k$ is given by a relation of type $f= Cm^{x}k^{y}$; where $C$ is a dimensionless quantity. The values of $x$ and $y$ will be:
1. $x=\frac{1}{2},~y= \frac{1}{2}$
2. $x=-\frac{1}{2},~y= -\frac{1}{2}$
3. $x=\frac{1}{2},~y= -\frac{1}{2}$
4. $x=-\frac{1}{2},~y= \frac{1}{2}$

The quantities $A$ and $B$ are related by the relation, $m= \frac{A}{B}$, where $m$ is the linear density and $A$ is the force. The dimensions of $B$ are of:

The velocity of water waves $v$ may depend upon their wavelength $\lambda$, the density of water $\rho$ and the acceleration due to gravity g. The method of dimensions gives the relation between these quantities as 

(1) $v^2\propto g{\lambda }^{-1}{\rho }^{-1}$

(2) $v^2\propto g\lambda \rho$

(3) $v^2\propto g\lambda$

(4) $v^2\propto g^{-1}{\lambda }^{-3}$

The equation of a wave is given by $Y=A\sin \omega (\frac{x}{v}-k)$ where $\omega$ is the angular velocity, x is length and $v$ is the linear velocity. The dimension of k is 

(1) LT

(2) T

(3) $T^{-1}$

(4) T²

The period of a body under SHM i.e. presented by $T=P^a{D}^b{S}^c$; where P is pressure, D is density and S is surface tension. The value of a, b and c are 

(1) $-\frac{3}{2},\frac{1}{2},1$

(2) $-1,-2,3$

(3) $\frac{1}{2},-\frac{3}{2},-\frac{1}{2}$

(4) $1,2,\frac{1}{3}$

The velocity of a freely falling body changes as $g^p{h}^q$ where g is acceleration due to gravity and h is the height. The values of p and q are 

(1) $1,\frac{1}{2}$

(2) $\frac{1}{2},\frac{{\displaystyle 1}}{2}$

(3) $\frac{1}{2},1$

(4) 1, 1

A small steel ball of radius $r$ is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta$. After some time the velocity of the ball attains a constant value known as terminal velocity $v_T$. The terminal velocity depends on $(\text{i})$ the mass of the ball $m$ $(\text{ii})$ $\eta$ $(\text{iii})$ $r$ and $(\text{iv})$ acceleration due to gravity $g$. Which of the following relations is dimensionally correct:

The quantity $X=\frac{{\epsilon }_{0}LV}{t}:{\epsilon }_0$ is the permittivity of free space, L is length, V is the potential difference and t is time. The dimensions of X are the same as that of  

(1) Resistance

(2) Charge

(3) Voltage

(4) Current

The dimensions of physical quantity X in the equation Force $=\frac{X}{\text{Density}}$ is given by 

(1) $M^1{L}^4{T}^{-2}$

(2) $M^2{L}^{-2}{T}^{-1}$

(3) $M^2{L}^{-2}{T}^{-2}$

(4) $M^1{L}^{-2}{T}^{-1}$